Final Exam Practice: fixed income -- answers

A. Binomial Option Pricing 30 points

Assume that the interest rate starts at 4% and in each period and either increases by 2% or decreases by 2% (from 4% up to 6% or down to 2% would be the first move). The risk-neutral probabilities of ups and downs are all 1/2.

What is the price now of a discount bond with face of $100 maturing one year from now?

     
This is a one-period discount bond with face of $100 and an interest rate
equal to the initial rate of 4%.  Therefore, the price is

 100
---- ~ $96.15
1.04

What is the price now of a discount bond with face of $100 maturing two years from now?

     
interest tree:

          8%
        / 
     6%
   /    \
4%        4%
   \    /
     2%
        \ 
          0%

discount bond price:

                100
              /
        94.34
      /       \
92.49           100
      \       /
        98.04
              \
                100

What is the price today of a two-year collar with a cap price of 5% and a floor price of 3%? The underlying notional is $1,000.

     
cash flows:
 
            30
         /
      10
   /     \
 0           0
   \     /
     -10
         \
           -30

price (pre-cash flow)

                    30
                 /
          24.151 
       /         \
-0.267               0
       \         /
         -24.706
                 \
                   -30

calculations:

10 + (30 + 0)/2/1.06 = 24.151
-10 + (0 - 30)/2/1.02 = -24.706

(24.151 - 24.706)/2/1.04 = -.267

B. Binomial Option Pricing 30 points

Assume that the interest rate starts at 6% and in each period and either increases by 2% or decreases by 2% (from 6% up to 8% or down to 4%). The risk-neutral probabilities of ups and downs are 1/2.

What is the price now of a discount bond with face of $100 maturing one year from now?
```
$94.3 = 100/1.06
 
```

What is the price now of a discount bond with face of $100 maturing two years from now?


quoted spot rates:

          10%
        /
     8%
   /    \
6%         6%
   \    /
     4%
        \
           2%

discount bond prices

               100
             /
        92.6
      /      \
$89.0          100
      \      /
        96.2
             \
               100

92.6 = 100/1.08, 96.2 = 100/1.04, 89.0 = 0.5 * (92.6 + 96.2)/1.06

What is the price today of an interest rate floor with a strike of 7% and two periods to maturity? The underlying notional is $100 (so the cash flow is 2 if the rate is 5%).
```
 
cash flows

        0
      /
    0
  /   \
0       1
  \   /
    3
      \
        5
 
Values

               0
             /
        0.46
      /      \
$2.99          1
      \      /
        5.88
             \
               5
```
0.46 = 0.5 * 1/1.08, 5.88 = 3 + 0.5*(1 + 5)/1.04, 2.99 = 0.5*(0.46 + 5.88)/1.06